#### Abstract

We use basic properties of superquadratic functions to obtain some new Hermite-Hadamard type inequalities via Riemann-Liouville fractional integrals. For superquadratic functions which are also convex, we get refinements of existing results.

#### 1. Introduction

Let be a convex function and with ; then is known as the Hermite-Hadamard inequality.

This remarkable result is well known in the literature as the Hermite-Hadamard inequality. Recently, the generalizations, refinements, and improvements of the classical Hermite-Hadamard inequality have been the subject of intensive research.

*Definition 1 (see [1]). *A function is superquadratic provided that for all there exists a constant such that
for all .

Theorem 2 (see [1]). *The inequality
**
holds for all probability measures and all nonnegative, -integrable functions , if and only if is superquadratic.*

The discrete version of the above theorem is also used in the sequel.

Lemma 3 (see [2]). *Suppose that is superquadratic. Let , , and let , where and . Then
*

Nonnegative superquadratic functions are much better behaved as we see next.

Lemma 4 (see [1]). *Let be a superquadratic function with as in Definition 1. Then one gets the following:*(1)*;*(2)*if , then whenever is differentiable at ;*(3)*if , then is convex and .*

The Hermite-Hadamard inequalities for superquadratic functions are established by Banic et al. in [3].

Theorem 5 (see [3]). *Let be a superquadratic function and ; then
*

It is remarkable that Sarikaya et al. [4] proved the following interesting inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals.

Theorem 6 (see [4]). *Let be a positive function with and . If is a convex function on , then the following inequalities for fractional integrals hold:
**
with .*

We remark that the symbols and denote the left-sided and right-sided Riemann-Liouville fractional integrals of the order with which are defined by respectively. Here, is the gamma function defined by .

Fractional integral operators are widely used to solve differential equations and integral equations. So a lot of work has been obtained on the theory and applications of fractional integral operators.

For more results concerning the fractional integral operators, we refer the reader to [5–10] and references cited therein.

In this paper, we establish some new Hermite-Hadamard type inequalities for superquadratic functions via Riemann-Liouville fractional integrals which refine the inequalities of (6) for superquadratic functions which are also convex.

#### 2. Main Results

From Lemma 3 for , we get for , , for the superquadratic function that hold, and therefore Let ; we get that Therefore, by replacing in (9) we get that

Theorem 7. *Let be a superquadratic integrable function on with . Then
**
with .*

*Proof. *By inequality (12), we get that
By the change of variable , we get
Therefore
We have completed the proof.

Corollary 8. *Putting in Theorem 7 gives
*

Let ; we get that Therefore, by replacing in (9) we get that

Theorem 9. *Let be a superquadratic integrable function on with . Then
**
with .*

*Proof. *By inequality (20), we get that
The proof is completed.

Corollary 10. *Choosing in Theorem 9, one has
*

Corollary 11. *Let be defined as in Theorem 7; one gets
*

Corollary 12. *Taking in Corollary 11, one obtains
*

#### 3. Conclusion

In this note, we obtain some new Hermite-Hadamard type inequalities for superquadratic functions via Riemann-Liouville fractional integrals. For superquadratic functions which are also convex, we get refinements of known results. The concept of superquadratic functions in several variables is introduced in [11]. An interesting topic is whether we can use the methods in this paper to establish the Hermite-Hadamard inequalities for superquadratic functions in several variables via Riemann-Liouville integrals.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work is supported by the Youth Project of Chongqing Three Gorges University of China (no. 13QN11).